J . Phys. Chem. 1993,97, 1231-1236
1231
Symmetry of Infinite Tubular Polymers: Application to Buckytubes D. J. Klein,’ W. A. Seitz, and T. C. Schmalz Department of Marine Sciences, Texas A& M University at Galveston, Galveston. Texas 77553- 1675 Received: October 12. I992
The geometric “line” symmetries possible for a long tubular polymer are identified and their irreducible representations developed, in a canonical format. Assumptions built around a subgroup of pure translations are avoided, while at the same time maintaining a “local” development that leads to a systematic presentation of band structure for electronic or vibrational structure computations. As an illustration of these techniques, they are applied to analytically solve the Hiickel model for all possible Ybuckytubes”,these being long tubular polymers composed entirely from a hexagonal net of carbon atoms. Of these tubular structures which are uniquely characterized, it is found that one out of three have spectra consistent with metallic behavior. Relationships to planar benzenoid polymers are also elucidated.
1. Introduction
There are many quasi- 1-dimensional molecular specieshaving macroscopic extent in only one direction while the extent in perpendicular directions is on the atomic scale. For regular polymers, composed from a single type of building block monomer unit, it is natural to model the system as infinite in the exceptional direction. Indeed this is common practice. Further all possible consequent line groups have formally been identified, e.g., in refs 1 and 2. But in treating molecular systems restriction is made to the case of a nonidentity subgroup of pure translations, as in refs 3-5. This typically carries over to treatment of irreducible representations3v6and band structure.’ In addition even for this restricted case these subjects are treated in a formally correct manner which nevertheless can disguise some “local” aspects. Other treatments* without the implicit assumption of nonidentity translational symmetry are more limited in the consideration of pssible symmetries and still disguise the same “local” aspects. The prototypical system under consideration may be represented as a set of equivalent helixes on the surface of a cylinder, each helix having internal repetitive structure. The circumstance is indicated in Figure 1 for the case of h = 3 helixes. There a particular location on one helix is indicated by a dark spot and nearby symmetry-equivalentlocationsare indicated by small opencircle spots. The number h of equivalent helixes we refer to as the helicity. The special case of h = 0 corresponds to a sequence of equivalent circles iterated along the length of the cylinder. Problems with the manifestation of “local” aspects in the group presentation and irreducible representations can be suggested from this figure in the case that the development is based either on the subgroup of pure translations or on the subgroupgenerated by an element with a minimal translational part. The puretranslation generator might carry the dark spot of Figure 1 to the distant asterisk to the right, or even much farther. The generator with the smallest nonzero translational component might have a large rotational component, e.g., carrying the dark spot to the asterisk on the cylinder’s back side, which again can be distant. An ideal circumstance for the application of these ideas is found with the now recently experimentallyrealized9buckyrubes, obtainable by wrapping a hexagonal net of carbon atoms onto a (long) tube. Indeed these species may be viewed as natural extensionsof the widely studied topic of carbon cages-theoretical work approaching the buckytube limit having been previously addressed to short tubesI0 with fullerene caps, finite tubes bent into a cycleIOJ1(i.e., a torus), and long tubes’* with fullerene caps. Recently too the infinite polymer” limit has been treated.
Figure 1. Helicity h = 3 tube. The solid dot indicates a reference site on one helix with nearby symmetry equivalent sites indicated by open dots. The asterisksindicate possible (somewhat distant) nearestlocations for sites equivalent to the solid-dot site by either a pure rotation or a pure translation.
After general symmetry considerationsin sections2-5, we deal with buckytubes in sections 6 and 7 and compare results for these species with a class of varying-widthbenzenoid polymers in d o n 8. The identification of all possible buckytubes composed from hexagonal nets is made here, with the idea that the seemingly ‘less symmetric”helical structures could possibly be synthetically favored, because of more simply conceived mechanisms of formation. That is, with a helical structure growth can be imagined to proceed via a single step, adding few-atom units one at a time to an evergrowing helix. But for a nonhelical (achiral) structure, local growth would be imagined to occur perhaps by three steps, say, beginning a new ring, continuing the ring, and closing the ring, so that the possibility of a “bottleneck” would be more likely. Conceptually the variety of possibilities might be conceived of as arising from the bending of different benzenoid polymer strips to wrap up on a cylinder with the top and bottom edges of a strip then fused together. Thence one might expect similarities between the properties of buckytubes and those predicted earlier for benzenoid polymer strips.&* Indeed we do find such similarities.
2. LiaeCroup The symmetry group of an extended quasi-one-dimensional object (in Euclidean 3-space.) is’s2a fine group G. Its elements arerepresentedas(correlated)pairs(RJt) ofa (properorimproper) rotation R and a translation t along the object axis a. In general some R may “reverse” the axis a in the sense that (RIO)(1It)(RlO)-’ = (ll-t) (2.1) That is, R might be a 2-fold rotation about an axis perpendicular to a,or R might be an improper rotation about a. The subset of all (Rlt) E G such that R does not so reverse a is seen to forms a subgroup G+,of index 2 or 1 in G. The subset of all (qt)E G such that R is a proper rotation also is seen to form a subgroup
0022-365419312097-1231%04.00/0 0 1993 American Chemical Societv
1232 The Journal of Physical Chemistry, Vol. 97, No. 6, 1993
Klein et al.
G+, again of index 1 or 2 in G. The intersection
Gf=G+nG+
k, (2.2)
+
+
=Tt
7
+
h-I
2xm kzm=O+L-1 L ’
so that Gf is commutative. This is the group that characterizes
TI
m,=O+p-l
P
2rm kll12 kh’ mll= 0 H
( ~ l t ) ( q u=) (mJt U) = (SR~U r ) = (Slu)(Rlt) (2.3)
the symmetry consequences of primary interest here. It is also of degree 1 or 2 in G+ and in G+. A characterization of the line group Gf may be made in terms of a set S of points generated via the action of the transformations of Gf from a single point 0. With o located at a characteristic distance r from the unique axis a of Gf, all the equivalent points lie on the surface of a radius r cylinder axially centered on a. Now each point q of the so-called’ orbit corresponds to a unique group element, namely, that which carries o to q. Choose a point q at a minimal distance from o as measured by a geodesic in the cylinder surface. The corresponding group element generates a cyclic group which however need not exhaust Gf. Thence from any remaining points not corresponding to this subgroup of Gf we pick a point q’as near as possible to 0 , again as measured by a geodesic in the cylinder surface. The group elements corresponding to q andq’may now be anticipated to beappropriate as generators for a (locally 2-dimensional) space as the surface of the cylinder appears on scales small compared to its radius r. Denote these two general group generators by T , and T I \ ,with T , being thcelement that is more purely rotational and 711the element that ismore purely translational. In thecasethat T , is the extreme of a pure rotation, it gives rise to a set of points equivalent to o all lying in a circular ring on the cylinder. Despite the possibility of a slight rotation of subsquent (marked) rings, we say this is the helicify h = 0 case. In the general case T , entails some (slight) translation whence T , generates a set of points equivalent to o all lying on a helix in the cylinder surface. Then repeated applications of 711 carries this helix into new helices. For our (discrete, atomic) polymer system the group Gf must have a finite number of orbit points per unit length of the cylinder, so that only a finite number of distinct such helices may arise. This number h is termed the system’s helicity. Then T~ must be the same as some power of T,, say
+-,k
P
has only (Rlt) with R a proper rotation around a. Thence for (Rlt) and (qu) in Gf
+
I-2*m,
(3.1)
and the irreducible representation matrix elements are
(3.2) This assertion may be verified by noting that the functions of (3.2) satisfy
6 ( k , A’, )a(kil,ki$a(k,k3 (3.3) which are the standard orthogonality conditions. There are a pair of related alternative forms for the presentation of symmetry results, based upon the elimination of T (via eq 2.4) in preference to either T , or ~ 1 1 . The associated new wave vector and irreducible representation for the case where T is replaced by ?llh are
K,
2um PL ’
I -
m = 0 +pL - 1
2um pK, kll=l h+ , mII=O+h-l h
Alternatively (for h # 0) we may replace T by
T,P
and obtain
2xm K l l = w m=O-+hL-I 2xm, k , E - + PJ ,
hK P
m,=O+p-l
(2.4)
Though T need not be a pure translation (nor do we assume that there necessarily are any other pure translations other than the identity), the numberp we term the period of our system. With the imposition of cyclic boundary conditions, one assumes TL = 1 (2.5) for some (very) large number L. The relations (2.4) and (2.5) provide our desired formal characterization of Gf. Any element of G : is uniquely specified as
(2.6) with a, 6, and c being nonnegative integers less than h, p, and L respectively. Indeed, a, 6, and c in (2.6) may be taken to identify what we call a reduced unit cell for the system. . l T p ~ ~
3. Irreducible Representations We desire a specification of all irrcduciblerepresentations(over the complex field) when the group elements are as in (2.6) with laws of multiplication as in (2.4) and (2.5). They are labelled by three wave vectors:
Either of these alternatives maintain the desired local features. In the h or p ~3 limit of large tubes, the pairs K,, k, or KII, k, may be viewed to approach the -continuum” pair k,, kyof the Euclidean plane. The distinctive feature of the k L , kll, k of the first paragraph is that the two local transformations are treated on an equal footing. For the large period and helicity limit (p, h -) the pair k,, kll may be viewed to approach the k,, k, pair of the Euclidean plane. --+
-
4. BandStructure
The line group of a polymer provides by way of its irreducible representationssymmetry-basedquantum numbers for electronic states. For the one-electron case symmetry orbitals may be constructed from localized orbitals. Let
(m;U,b,C)I T ~ T f T c l m ; O , O , O ) (4.1) denote the mth orbital of reduced unit cell a, 6, c. The symmetry orbitals are
The Journal of Physical Chemistry, Vol. 97, No. 6,1993 1233
Symmetry of Infinite Tubular Polymers
H
If the Im;a,b,c)in different cells are orthogonal, then
(m;k,,k,,,klmlk’,,k~,k? = b(k,,k’,)b(kll,kil)6(k,k3S,,
X
(4.3) with the overlap Smml, independent of k l , kll, k and the same as arises for the localized orbitals of (4.1). The representation of a (symmetric) one-electron Hamiltonian is block-diagonalizedon the basis of symmetryorbitals. In accord with our “local” presentation of Gf such a Hamiltonian is conveniently resolved as (4.4)
Figure 2. (a) A 5-hexagon portion of a polyacene chain; (b and c) two
where H(bl,611,6) is that part of H carrying localized orbitals of reduced unit cell a, b, c to the cell u b l , b + 61, c + 6. Because of our locality arrangements 61, 611,and 6 should all be limited to small magnitudes, most simply 0 or ( 7 )1. The matrix elements then are
+
possible ways considered to initiate three polyacene chains from a given reference hexagon. /----------
-
7- 7,
(m;k,,kll,kJH(S,,bI,,b)ln;k’,,k;l’k? = m , , k ’ J 6 ( k , , k y x ~(k,k?e’(kl~l+kl1611+kd) Bm,(Sl,611,6) where the cell-independent local integrals are
(4.5)
Bm,(S,,Sll,S) I (m;a+b,,b+bll,c+bl~n;a,b,c) (4.6) with a + al, b 611,c 6 representing additions modulo p, h, L,respectively. This then gives rise to small matrices (s X s, if s is the number of orbitals per unit cell), which dependanalytically on the (near) continuous variable k, thereby giving smooth eigenvalues, and bands. Following the comments near the end of section 3, one may view kil or kl as labeling a discrete set of one-dimensional cross-sections from a two-dimensional band surface. Similar consequences arise for vibrational treatments. The symmetry-adaptedphonon modes are much as in (4.2) represented in terms of local atomic vibrations within the reduced unit cells. The block-diagonalization of the vibrational Hamiltonian (and matrices) then follows the pattern evident in (4.5).
+
+
5. Comment8
The results embodied in eqs 2.4 and 2.5, in eqs 3.1 and 3.2, and in eqs 4.1 and 4.5 are our primary results for the line-group presentation,theirreduciblereprcsentations,and theconsequences for band structure, respectively. There are however other possible presentations and subsequent developments. The presentation could be based upon group elements with the smallest translation components: after factoring out of Gf the subgroup of pure rotations, a remnant element f with the smallest (nonzero) translation would generate all of Gf that remains. As indicated toward the end of section 1, f can effect a “nonlocal” transformation which then would be manifested, e.&, the consequent wave vector does not so simply correlate with the k,, ky of the Euclidean plane and there can arise violent oscillations in the band-structure prcsentation. Notably no assumption as to nonidentity pure translations has been made. Indeed there seems little warrant for this assumption for isolated polymers, such as seems appropriate for the experimentally realized9 ”buckytubes”,which also seem to occur with nonzero helicity
.
6. Pwible Buckybbe8
For our basic characterization we imagine each buckytube to becomprised from one or more regular unit-hexagon-width strips fused together in the appropriate fashion. Such a prototypical polyacene strip is shown in Figure 2a. Then from any representative hexagon on the cylindrical surface of a buckytube we
(a1 (b) Figure 3. Indications of the manner of subsequent self-intersection of three polyacene chains radiating from a single hexagon.
imagine three such polyacene strip radiating away, at 120° angles with respect to one another, as indicated in Figure 2b. There is a second choice of three directions from our focal hexagon, as indicated in Figure 2c, but this choice gives equivalent results in the followingdevelopment. Imagine each of the three polyacene chains is continued until intersection with some other part of the three radiating polyacene chains. Either the first point of intersection is in the original focal hexagon, as in Figure 3a, or else it is in another hexagon, as in Figute 3b. Thence in every case a cycle of hexagons is found, each cycle being characterized by two nonnegative numbers corresponding to the numbers of hexagons in the two intersecting polyacene arms from the center of the focal hexagon to the center of the intersecting hexagon. The larger of these two numbers t+ is termed the twist and the smaller 1- is termed the countertwist of the buckytube. The twist and countertwist labels provide our basic characterization of buckytubes. Fragments of countertwist t- = 0, 1, and 2 tubes are indicated in parts a+ of Figure 4, respectively. Evidently a twist without any countertwist(Le., t- = 0) comsponds to a pure rotational symmetry of the tube. Also a twist and countertwistof the same size (i.e., t+ = t-) entails a pure rotational symmetry, such a circumstance being indicated in Figure 4d. Hence the twist defect t6 t+ - t- may be viewed as a relevant characterization of buckytubes, while the twist sum t, = t+ tmay be viewed as a type of “period” or “circumference” (along the two intersecting polyacene strips). A case with t( = 1 is indicated in Figure 4e. Evidently the necessary and sufficient condition for a (chiral) helical structure (at least if the hexagonal rings are of maximal symmetry) is t f g # 0. In such cases an additional point of characterization .pight be to assign a “parity” or - to each helix as the (longer) $+-length polyacene strip winds like a right- or left-handed sbrew thread. But such a distinction is irrelevant for the band structure of the next section and for many other properties as well. A further informative picture arises for buckytubes if we view the polyacene strip along the t+-twist direction to be extended in both directions along the wholelength of the strip. If r- # 0, this strip forms a helix and t- counts the number of such (equivalent) helixes that are fused together to form the buckytube. Thence t- may be viewed as the “helicity” of section 2. If t- = 0, then a ring is obtained and an infinite sequence of them is fused to form the buckytube. Alternatively pdyacene strip along the 1-
+
+
1234 The Journal of Physical Chemistry, Vol. 97, No. 6,1993
Klein et al.
b
a
C
e
d
Firmn 4. Structures for small mrtions of countertwist t- = 0.. 1.. and 2 buckytubes in (a-c) respectively. (d and e) Small portions of tubes with twist d e b
ra = 0 and
1.
(a)
@+t-
(b)
a wme vector ranging (essentially continuously) from I to +r, and k being a quasi-discrete quantity taking values
k= ,J I,
i
t+
Figure 5. (a) Typical reduced unit cell enclosed within dotted lines; also shown are the directions taken to make the t+ and t- counts of the text. (b) Labels for the bonds incident in a reduced unit cell.
direction (for t- # 0) could be considered, whence t+ would count helixes, generally of a different pitch than before. Notably in any case each of the polyacenic components are equivalent, and within any (possibly cyclic) polyacenic strip any pair of sites on a fusion edge within the strip is equivalent to every other such pair. That is with the assumption of maximal (geometrical) buckytube symmetry, any adjacent pair of sites is equivalent through a screw translation to every other like-oriented adjacent pair. 7. Bandstructure
For all the buckytubes identified there are (diamond-shaped), reduced unit cells with but two sites, as indicated in Figure 5. We imagine on each site a single d i k e orbital oriented normal to the tube surface. In general the nearest-neighbor Hiickel (resonance) integrals along bonds in different directions will be different, say &, Bb, and 8, (in expected order of decreasing magnitude, as judged from the differing extents of curvature of the tube surface along the different directions,as indicated in Figure Sb). Thence 2 X 2 Hiickel matrices result, following symmetry adaptation. Since the diagonal elements may be taken as 0, only one of the two remaining (complex conjugate) off-diagonal elements need be identified:
H,,= 6,+ ~ ~ e+-BOe+"& ' ~
(7.1) Here k and K identify the (irreducible) symmetries, with K being
[
2 r m / t + , t- = 0 2rmlt-
+ t+K/t-,
t- #
0
(7.2)
where m ranges over t+ and t- successive integers. Thence the Hiickel model eigenvalues are
+
+
ct(m,K) = Ibc BOeikl (7.3) where it is understood m and k correspond as in (7.2). The energy bands with K varying over a range of 2 r are each labeled by the discrete variables & and m. Because of "alternancyn the bands are symmetric about t = 0. A key point of interest is the band (or HOMO-LUMO) gap-particularly whether or not it is 0. In the simplest case one approximates the three different (resonance) integrals by the same 8. Then the necessary and sufficient condition for a zero bandgap is 1 + e-iK + eik =
0
(7.4)
which in turn may be seen to occur if and only if
k = K = f 2 r / 3 , mod 2% (7.5) Checking the or -possibilities of (7.5) against the twodifferent expressions, we finally obtain
+
-
0 band gap t6 divisible by 3 (7.6) Further in the zero-band-gap case the crossing occurs at K = f 2 r / 3 between the bands with m = f r 6 / 3 . Thence in such a caw there is the possibility of a Peierls distortionl7J* and consequent solitonic behavior.19 Upon varying 8., Bb 6, the crossings should move away from K = + 2 r / 3 , but to a lesser extent the larger a tube's circumference (as measured by ts).The resultant distortively produced band gaps should be less than in polya~ctylene,~~J~ approaching the 0 limit of graphite as rs increases. Thence at ordinary temperatures for larger tubes no distortion may occur and the polymers may remain metallic. This prediction extends that of Mintmire et al.13 for the 16 = 0 caw. The possibility for superconductionis intriguing too especially in light of trends20 reported for doped 'fullerite".
The Journal of Physical Chemistry, Vol. 97, No. 6,1993 1235 0
2
0.4-
a
0.3-
.. . *
0.2-
0.1-
t
d
I
l/t,
0.1
0.05
0.15
1
3
0.2
(b)
0.2
Fipurel. Two benzenoid strips which when wrapptdontoa tube preserve their nonbonding MOs, one of which is indicated here with solid and open circles corresponding to positive and negative amplitudes, while uncircled vertices correspond to 0 amplitudes.
.*
0.1
c
I
l/t,
0.1
0.05
0.15
0.2
(Band Gap@ 0.41 C
O 0 .S23 l
0.11
.*.
.. .‘
I
l/t, 0.1 0.15 0.2 Figure 6. Plots of bandgap versus 1/r, for the r- = 0, 1, 2 classes of buckytubes.
0.05
The nonzero-bandgap buckytubes (with ta f 1 divisible by 3) areof interest too. Of course the bandgaps should again approach 0 as the ts graphitic limit is approached. This is indicated in Figure 6 for uniform-@tubes of different countertwists t- = 0, 1, 2. For each of these three considered classes the plots of bandgap versus 1/ t , reveal two curves, one for ta + 1 divisible by 3 and the other for ta - 1 divisible by 3.
-
8. Discdon
The modulo-3 pattern of 0-bandgap buckytuba is remini~cent’~ of module3 patterns for related benzenoid strips. These earlier patterns first noted in a resonance-theoretic context seem15 to correspond to HUckel-theoretic 0-bandgaps. There are two classes of polymers of increasing widths w such that every third width has16*21a 0-bandgap. The 3-hexagon-widestrip of one class16is indicated in Figure 7a and the 9/2-hexagon wide strip of the other classZi is indicated in Figure 7b. One nonbonding MO for each of these strips is also indicated, with solid and hollow circles indicating oppositely signed nonbonding MO coefficients. The pattern of these coefficients is readily seen to be extendable so long as the strip width increases by any multiple of 3 so that corresponding 3n-hexagon wide and 3(n i/2)-hexagon wide strips also have nonbonding MOs and thence 0-bandgap. But now each of the strip shown can be bent onto a cylinder to form buckytubes. The top and bottom edges of the strips are to be brought into coincidence, but in general this may be done in more than one way, so that in Figure 6a the protruding hexagon indicated by an arrow on the bottom of the strip could be “fused” intoeitheroftheslotslabeledOor2at the topofthestrip, thereby giving t- = 0 (and t+ = 3) or f- = 2 (and t+ = 5 ) buckytubes. Similarly the strip in Figure 6b leads to 1- = 1 (and t+ = 4) or t- = 3 (and t+ = 6) buckytubes. Significantly one sees that the
+
nonbonding MOs with exactly the same coefficientpatterns persist for these buckytubes. Thence the module3 pattern (of eq 7.6) for buckytubes is fundamentally related to that for theprwiousl6J1 familiesof benzenoid strips,the nonbonding MOs being intimately related (if not the “same”). In conclusion we have elucidated not only the conceivable buckytube structures but also their HUckel band spectra and the structural character of and criteria for nonbonding MOs and the associated metallic 0-bandgap. Since submission of this work in March, several theoretical papers on buckytubes have appeared.22-26The most significant overlap with the present work is found in refs 22-25, which also make HUckel-type tight-binding treatments. The first three of these identify the full class of possible buckytube structures (in different manners), each reports the results of numerical computations on a subset of less than a dozen cases, and each identifies suggested sufficient conditions for zero-bandgap buckytubes. References 24 and 25 develop analytic results for the nonhelical cases with t-26 = 0. However, we believe our present work to be the first comprehensiveanalytic treatment of the problem and the first to emphasize the correspondences to strips, as in section 8 here. Also the general group-theoretic results of sections 3 and 4 should find further use.
Acknowledgment is made for partial support of this rcscarch to The Welch Foundation of Houston, Texas and to the donors of The Petroleum Research Fund, administered by the American Chemical Society.
References and Notes (1) Shubinikov, A. V.; Koptik, V. A. Symmerry in Science and Art; Plenum Press: New York, 1974. (2) Lockwood,E. H.;Macmillan, R. H. GeomerricSymmerry;Cambridge University Press: London, 1978. (3) (a) Tobin, M. C. J. Mol. Specrrosc. 1959, I,349-358. (b) Zbinden, R. Infrared Spectroscopy of High Polymers; Academic Press: New York, 1966. (4) Vainshtein, B. K. Dif/rcrcrion of X-rays by Chain Molecules; Elsevier: Amsterdam, 1966. ( 5 ) VujiSC, M.; BozOviC, I. B.; Herbut, F. J . Phys. 1977, AIO, 1271. (6) BohviC, I. B.; Vujiait, M.; Herbut, F. J . Phys. A 1978, 11, 2133. (7) (a) Del Re, 0.; Ladik, J.; Biczo, G . Phys. Rev. 1967, ISS,997. (b) Botov~C,I. B. Phys. Rev. B 1984, 29, 6586. (8) Imamura, A. J . Chem. Phys. 1970.52, 3168. (9) (a) Iijima, S. Nature 1991, 351, 56. (b) Ebbesen, T. W.; Ajayan, P. M. Nature 1992, 358, 220. (10) Schmalz. T. G.; Seitz, W. A.; Klein, D. J.; Hite, 0.E. 1.Am. Chem. Soc. 1988, 110, 1113. (11) Tosi, R.; Cyvin, S. J. J . Marh. Chem., submitted. (12) (a) Fowler, P. W. J . Chem. Soc.,Faraday Trans. 1990,86,2073. (b) Dresselhaus, M. S.; Dresselhaus, G.;Saito, R. Phys. Rev. E 1992,45,6234. (c) Harigaya, K. Phys. Rev. E 1992, I S , 12071.
1236 The Journal of Physical Chemistry, Vol. 97, No. 6, 1993 (13) Mintmire, J. W.; Dunlap, B. 1.; White, C. T. Phys. Reu. Lett. 1992, 68, 631. (14) (a) Klein, D. J.; Schmalz, T. G.;Hite, G.E.; Metropoulos, A.; Seitz, W. A. Chem. Phys. Lett. 1985. 120, 367. (b) Klein, D. J.; Hite, G. E.; Seitz, W. A.; Schmalz, T. G. Theor. Chim. Acta 1986, 69, 409. (c) Seitz. W. A.;
Hite, G. E.; Schmalz, T. G.;Klein, D. J. Graph Theory and Topology in Chemistry; King, R. B., Rouvray, D. H., Eds.; Elsevier: Amsterdam, 1987; pp 458465. (15) (a) Klein, D. J.; Schmalz, T. G.; Seitz, W. A.; Hite, G. E. Int. J . Quantum Chem. 1986, S19.707. (b) Seitz, W. A.; Schmalz, T. G.ValenceBond Theory and Chemical Structure; Klein, D. J., TrinajstiC, N., Us.; Elsevier: Amsterdam, 1990; pp 525-552. (16) Klein, D. J. Rep. Mol. Theory 1990. I , 91. (17) Peierls, R. E. Quantum Theory of Solids; Oxford University Press: London, 1955; p 108. (18) Longuet-Higgins, H. C.; Salem, L. Proc. R. Soc. (London) 1959, A251, 172. (19) Su, W. P.; Schrieffer, J. R.; Heegcr, A. J. Phys. Rev. Lett. 1979,42, 1698; Phys. Reu. B 1980, 22, 2099.
Klein et al. (20) (a) Fleming, R. M.; Ramirez, A. P.; Rosseinsky, M. J.; Murphy, D. W.; Haddon, R. C.; Zahurak, S.M.; Makhija, A. V.Nature 1991,352,787. (b) Iqbal, Z.; Baughman, R. H.; Ramakrishna, B. L.; Khare, S.;Murthy, N. S.;Bornemann, H.J.; Morris, D. E. Nature 1991, 254, 826. (21) Bradburn, M.; Coulson, C. A.; Rushbrooke, G. S. Proc. R . SOC. Edinburgh A 1948, 62, 336. (22) Hamada, N.; Sawada, S.; Oshiyama, A. Phys. Rev. Lett. 1992,68, 1579. (23) Tanaka, K.; Okahara, K.; Okada, M.; Yamabe, T. Chem. Phys. Lett. 1992, 191, 469. (24) Saito, R.; Fujita, M.; Dresselhaus, G.;Dresselhaus, M. S.Phys. Reu. B 1992, 46, 1804. (25) Gao, Y.D.; Herndon, W. C. Mol. Phys., in press. (26) (a) Robertson, D. H.; Brenner, D. W.; Mintmire, J. W. Phys. Reo. E 1992, 45, 12592. (b) Adams, G.B.; Sankey, 0. F.; Page, J. B.; OKeefe, M.; Drabold, D. A. Science 1992, 256, 1792. (c) Jishi, R. A.; Dresselhaus, M. S . Phys. Rev. B 1992,45, 11305. (d) Mintmire, J. W.; Robetson, D. H.; Dunlap, B. 1.; Mowrey, R. C.; Brenner, D. W.; White, C. T. Mater. Res. Soc. Symp. Proc. 1992, 247, 339. (e) Endo, M.; Kroto, H. W. J. Phys. Chem. 1992, 96, 694 1.
